A-site cation influence on the conduction band of lead bromide perovskites

Hot carrier solar cells hold promise for exceeding the Shockley-Queisser limit. Slow hot carrier cooling is one of the most intriguing properties of lead halide perovskites and distinguishes this class of materials from competing materials used in solar cells. Here we use the element selectivity of high-resolution X-ray spectroscopy and density functional theory to uncover a previously hidden feature in the conduction band states, the σ-π energy splitting, and find that it is strongly influenced by the strength of electronic coupling between the A-cation and bromide-lead sublattice. Our finding provides an alternative mechanism to the commonly discussed polaronic screening and hot phonon bottleneck carrier cooling mechanisms. Our work emphasizes the optoelectronic role of the A-cation, provides a comprehensive view of A-cation effects in the crystal and electronic structures, and outlines a broadly applicable spectroscopic approach for assessing the impact of chemical alterations of the A-cation on perovskite electronic structure.

Supplementary Fig. 10. Photoelectron survey/overview spectra recorded from single crystals of FAPB, MAPB and CsPB. The spectra were recorded from uncleaved single crystal surfaces.
Supplementary Fig. 11. Beam damage checks with photoelectron core level measurements. Nitrogen 1s (FAPB, MAPB) or cesium 3d (CsPB) and lead 4f (all) core level photoelectron spectra recorded from a,d FAPB, b,e MAPB and c,f CsPB. For each compound, all core level spectra were recorded from the same spot of the single crystal sample. The time(s) associated with iteration(s) ≥ 2 represents the elapsed time since the first iteration was recorded. During this time, the same spot was continuously exposed to the X-ray beam. The y-axis label for all plots is normalized intensity in arbitrary units.   Fig. 1b-e) show noticeable differences for incident photon energies below ~13042 eV. We observe diagonal constant energy loss features for all of the compounds, where the band of loss features is energetically narrower for PbBr2 and wider for the APB compounds, and the edges of this band are sharper for FAPB versus CsPB. The constant energy loss features for the APB compounds intersect the HERFD-XAS line cut at ~13036 and ~13041 eV, giving rise to two prominent features in the Pb L3 HERFD-XAS spectrum. These HERFD-XAS features have previously been assigned to excitations from Pb 2p3/2 to Pb 6d states hybridized with Br states, where the energy split between the two features originates from octahedral crystal/ligand field splitting 6 .
The improvement in intrinsic energy resolution yielded by Pb L3 HERFD-XAS (~2.5 eV), as compared to TFY-XAS (~6.1 eV), is substantial and crucial for detecting differences in spectral features. Supplementary Fig. 1f visualizes this point. This demonstrates the sensitivity of HERFD-XAS to chemical state and/or crystal structure changes as demonstrated by past Pb L3,1 HERFD-XAS studies of lead(II) compounds with different lead-ligand coordination and local structure 7 .
Supplementary Note 2. Comparison between experimental and calculated Br K XAS spectra of FAPB, MAPB and CsPB.
In all cases, Fig. 2c-e, we observe that the heights of the main-edge profile (up to ~13476 eV) relative to the post-edge region (above ~13476 eV) appear smaller in the calculated spectra than in experiment.
We also see in the calculations that the post-edge in each system, above ~13476 eV, is built up from two bands with σ-symmetry and a central π-symmetry band, but that is not resolved in the experiment. The small π-bonding contributions at similar excitation energies to the σ-bonding contributions in the mainedge, from ~13464 to ~13469 eV, may appear because the Pb-Br-Pb is bent.
We examine the computed peak positions of the σ main-edge feature, the main constituent of the absorption onset, and find a trend where the peak position increases, going from FAPB (13468.2 eV) to MAPB (13468.3 eV) to CsPB (13468.4 eV). The A-cation-influenced trend in the σ-π splitting, one of the key findings in our work, is unaffected by the Br K absorption onset (where our calculations show 0.1 eV relative changes) as the splitting spans several eV of the conduction band region. Our XAS calculations do correctly reproduce the absorption onset trend (increases in this order: CsPB  MAPB  FAPB). While the relative differences between MAPB and FAPB (experiment: 0.5 eV, calculation: 0.1 eV) were not fully captured, the relative differences between CsPB and MAPB (experiment: 0.1 eV, calculation: 0.1 eV) were fully captured. Overall, the quantitative σ-π splitting trend and the qualitative absorption onset trend hold.
The transition potential DFT methodology has been demonstrated to give useful support to the analysis of experimental data in numerous applications: (i) different element K-edge XAS and (ii) molecular and condensed phases. It is an approximate method, but has the advantage that it often gives semi-quantitative results combined with an excellent scaling to large systems. Ekimova et al. 8 used a similar methodology to investigate hydrogen-bonded solutes in an environment of 63 water molecules, and Hou et al. 9 used a similar methodology to investigate nitrogen-doped graphene. Dalpian et al. 10 have shown that it is essential to use a large supercell for accurately modeling the ground-state crystal structures of HaPs; this necessitates the use of methods with excellent scaling characteristics.
Supplementary Note 3. Potential DFT-related limitations with the calculation of the conduction band electronic structure.
The common exchange-correlation functionals used in DFT are known to yield underestimated bandgaps. However, the occupied and unoccupied parts of the computed DOS are generally in good agreement with experimental photoelectron spectra, stretched by a few percent 11 . Within the conduction band region, systematic DFT underestimation of a few percent or ~ 0.1 eV is not expected to affect the σ-π splitting trend as the magnitude of the splitting is ~ 4 eV.
DFT, using the same exchange-correlation functional as what we have used (Perdew-Burke-Ernzerhof (PBE) generalized gradient approximation (GGA)), has been demonstrated to be reasonably accurate at modeling the conduction band electronic structure of MAPI, a closely related compound 12 . While there is overall agreement between the calculated conduction band dispersions and the measured bands (obtained via angle-resolved inverse photoelectron spectroscopy) over a ~3 eV region of the conduction band, the agreement is not perfect at the 0.1 eV order-of-magnitude energy scale. We also notice that the good performance of the PBE approximation for lead-based halide perovskites is related to a favorable error cancellation (e.g. Mosconi et al. 13 , Das et al. 14 ). It has been reported in the literature that the performance of standard GGA is good for a number of response properties (e.g. Drisdell et al. 6 ).
Supplementary Note 4. Analysis of Br K XES spectra.
Bromine K XES provides a complementary bromine p-state-selective measurement of the occupied states to PES (but with significant (Br 1s related) core hole lifetime and instrumental broadenings) 15 . To investigate the profile of the Br 4p PDOS near the VBM (~13462 to ~13472 eV), the VtC main line in the XES spectra of FAPB, MAPB and CsPB was fitted with a Voigt peak. The range of the fit covers 13443 -13481 eV. The fitted Voigt FWHM parameters are comparable (4.1-4.2 eV).
From Fig. 4 (center inset), we observe that the relative VtC intensity of FAPB is higher than MAPB/CsPB given normalized Kβ1,3 intensities. This shows the Br 4p electron occupancy is higher for FAPB and suggests the Br-Pb bond for FAPB is more ionic relative to MAPB/CsPB. Higher ionicity (or alternatively, lower covalency) means less Br 4p electron sharing and hence higher orbital occupancy. To verify this finding, we examine the Kβ1 and VtC transition energies. From an electrostatic standpoint, given the same formal oxidation state, the Br Kβ1 transition energy is expected to be lower for a more ionic Brcompound due to better shielding of the nucleus by the higher valence electron density. This leads to a lower effective nuclear charge and consequently to a lower excitation energy for a core electron. X-ray emission spectroscopy studies of I --containing compounds show this trend with I Lγ emission 16  Our Br K XAS investigation shows the same trend, with an increasing absorption onset trend of FAPB→MAPB→CsPB, indicating that the chemical shift applies to all occupied and unoccupied energy levels. The XAS and ground-state σ-π splittings both show a decreasing trend of FAPB→MAPB→CsPB, hence we find a correlation between A-cation and Br-Pb sublattice electronic coupling strength and Br-Pb bond ionicity. The Br-Pb bond ionicity trend is the inverse of the Kβ1 transition energy trend and increases in this order: CsPB→MAPB→FAPB. Using frequency-dependent dielectric measurements, others have reported a trend where the Br-Pb bond in CsPB is more ionic relative to MAPB, contrary to our finding [19][20][21] . Experimental non-idealities such as electrical contacts may influence the dielectric measurements.
We notice however that the differences in Br-Pb bond ionicity are not captured in any charge analysis of the ground-state DFT calculations, despite the good agreement of these models in the analysis of the unoccupied states.
Supplementary Note 5. Modeling valence band photoelectron spectra with calculated ground-state DOS.
We utilize PES in its hard X-ray form, HAXPES, both to enhance the bulk sensitivity of the measurement and to emphasize the lead and bromide spectral contributions. Given the 4000 eV excitation used, the photo-ionization cross-sections of the Pb and Br states are 2-3 orders of magnitude higher than the cross-sections of carbon, hydrogen and nitrogen (Supplementary Table 3). The relative photoionization cross-sections for Pb 6s (~0.6), Pb 6p (~0.4) and Br 4p (1.0) states at hv = 4 keV are comparable. Groundstate element-and orbital-projected DOS calculations, derived from the same underlying AIMD simulations used to calculate the Br K XAS spectra (Fig. 2c-e) and Br PDOS (Fig. 3) and Gaussianbroadened by 0.3 eV to match the experimental energy resolution, are compared to HAXPES valence band spectra in Supplementary Fig. 6. A Fermi edge/step feature is visible in the valence band spectrum of MAPB; the origin of the metallic states is likely the silver contamination, as mentioned in Supplementary Note 7. We note two observations from the comparison. First, most of the features in the valence band spectra of FAPB, MAPB and CsPB, extending from the Fermi energy to the Pb 5d shallow core level can be approximately accounted for with the lead and bromine PDOS, except for the Cs 5p semi-core level in the case of CsPB. This is consistent with the relatively low photo-ionization cross-sections of C 2p, N 2p and Cs 6s states at hv = 4 keV. Second, the photoelectron spectra of FAPB ( Supplementary Fig. 6a) and CsPB ( Supplementary Fig. 6c), which are the endmembers of our set in terms of A-cation electronic coupling strength, show differences in the VBM DOS. This observation indicates that the A-cation replacement modifies the degree of Pb 6s and/or 6p hybridization with the Br 4p states, which may be relevant for optoelectronic functionality. Lead 6p states predominate in the conduction band and the probabilities of visible optical absorption/radiation events, relevant for solar cells and LED's, may increase (due to s↔p dipole selection rules for optical transitions) as the Pb 6s contribution grows at the VBM. The degree of Rashba spin-splitting at the VBM, if present, may be modulated by changes in Pb 6p contributions to the VBM as such states exhibit spin-orbit coupling 22 .
Supplementary Note 6. Connection between A-cation electronic coupling strength and crystal structure.
We examine the crystal structure for manifestation(s) of A-cation electronic coupling, in terms of correlation(s) with the σ-π splitting, chemical shift, etc. The crystallographic parameters obtained from XRD measurements are summarized in Supplementary Table 2; these are comparable to literature values and show that the structural phases of CsPB, MAPB and FAPB are orthorhombic, cubic and cubic at ambient conditions (room temperature, 1 atm pressure).
The Pb L3 HERFD-XAS spectra for the three APB compounds and PbBr2 are shown in Supplementary Fig.  1f. We observe two prominent features in all of the spectra: a rising-edge feature at ~13036 eV and a main-edge feature at ~13041 eV. Drisdell et al. 6 have assigned the origin of these features to hybridized Br-Pb 6d crystal field splitting. Between the three APB compounds, the sharpness of the two features has an apparent dependence on the type of A-cation and reveals differences in unit cell symmetry. The sharpness of the crystal field splitting features qualitatively increases in this order: CsPB→MAPB→FAPB. This indicates that the unit cell of FAPB shows the highest cubicity. The lower symmetry of PbBr2 and CsPB leads to an additional broadening and smearing of the spectral structures compared to MAPB/FAPB.
We inspect several structural descriptors: (a) Br-Pb-Br bond angle, (b) Br-Pb bond distance, (c) Pb-Br-Pb bond angle and (d) Pb-Pb distance, obtained from AIMD simulations for correlation(s). The corresponding plots are displayed in Supplementary Fig. 7. Since the bond angle distributions are asymmetric, we use a center-of-gravity analysis to quantify the mean values. The Br-Pb-Br bond angle shows a systematic trend towards higher bond angle (CsPB→MAPB→FAPB) though the differences are small (mean bond angle for FAPB is ~0.8% greater than for CsPB) and the distributions of Br-Pb bond distance are comparable for the three APB compounds. This suggests the internal structure of the PbBr6 octahedral unit is weakly affected by A-cation electronic coupling, though the effect on the electronic structure could still be substantial. On the other hand, the Pb-Br-Pb bond angle distribution shows a larger difference (i.e. the mean angle for FAPB is ~4.5% larger than the angle for CsPB) and a systematic CsPB→MAPB→FAPB trend towards higher angle. A larger Pb-Br-Pb bond angle signifies higher cubicity and unit cell symmetry, which is consistent with the Pb L3 HERFD-XAS feature sharpness trend. The Pb-Br-Pb bond angle quantifies the degree of cooperative tilting of the PbBr6 octahedra. The larger Pb-Pb distance for FAPB is consistent with the larger Pb-Br-Pb bond angle. From a purely geometrical standpoint, the Pb-Br-Pb bond angle is affected the most by the A-cation. We observe a positive correlation between the Pb-Br-Pb tilt angle and the XAS σ-π splitting, but are unable to validate the tilt as the mechanism responsible for the σ-π splitting using individual configurations. Hence, we deduce that the tilt angle and the σ-π splitting are not mechanistically related but both are consequences of the A-cation electronic coupling strength.
We examine the GTF, a structural descriptor which is applicable to all ABX3 perovskites (where X = halide, oxide, fluoride, etc.), for a complementary view of the crystal structure 23 . The GTF was introduced nearly a century ago and is familiar to the oxide/fluoride/halide/etc. perovskite communities 23,24  and CsPB : FAPB ratios of 0.83 (experiment) and 0.83 (calculated). Experiment refers to the relative Br K XAS main-edge width and calculated refers to the σ-π splitting derived from the calculated Br K XAS spectra. A positive and potentially linear correlation exists between relative GTF and relative σ-π splitting. Since we have found no evidence for a mechanistic relationship between the tilt angle (represented by the GTF) and the σ-π splitting (which we found to be affected by the A-cation electronic coupling strength), the apparent linear correlation between the σ-π splitting and GTF implies that the underlying mechanism which is responsible for both is the strength of electronic coupling between the A-cation and the bromide-lead sublattice.
Supplementary Note 7. Analysis of HAXPES core level spectra and potential beam damage effects.
The survey spectra were checked to ensure the expected elements are present ( Supplementary Fig. 10).
Since the crystals were uncleaved, C 1s and O 1s signals originating from surface contamination are observed. In addition, some unexpected silver contamination, likely originating from the silver epoxy used to bond the crystals to the sample plate, was observed from the surface region of MAPB. In general, all of the expected elements (e.g. bromine, lead, etc.) are present.
To assess the chemical integrity of the X-ray-irradiated surfaces, sequential measurements of core level spectra (Pb 4f for the bromide-lead sub-lattice, Cs 3d or N 1s for the A-cation) were monitored for changes ( Supplementary Fig. 11). For each sample, all core level spectra were recorded from the same spot on the crystal. The valence band spectra were typically recorded in between the first and last Pb 4f measurements. Negligible changes are observed from the sequential measurements of core level spectra, indicating that the recorded valence band spectra presented in the following section are not substantially affected by beam-induced chemical changes.
Supplementary Discussion 1. Potential influence of organic A-cation rotations and oscillations on the σ-π splitting and hot carrier cooling rate.
The timescales of organic cation rotation/oscillation in MAPB and FAPB have been found, via IR spectroscopy, solid-state NMR, etc. to be in the range of 0.3 -2 ps and 0.1 -2 ps, respectively 28 . Our HERFD-XAS measurements were recorded with a 1 second integration/accumulation time per energy point, whereas the XAS process is ultrafast. Likewise, the density of states and XA spectrum simulations are based on instantaneous snapshots from ab initio molecular dynamics simulations. Consequently, we are sampling the time-averaged electronic structure over virtually instantaneous configurations, and find the σ-π splitting to be a time-averaged feature in the conduction band. This is analogous to sampling the time-averaged crystal structure of HaPs with bulk X-ray diffraction, which shows highly crystalline long-range order in spite of the picosecond-timescale structural disorder 29,30 .
In the theoretical simulations, we find a positive correlation between the strength of N-H … Br hydrogen bonding and the time-averaged magnitude of the σ-π splitting: CsPB (no H-bonding, σ-πcalculated = 3.5 eV)  MAPB (H-Br bond distance ~2.47 Å, σ-πcalculated = 4.0 eV)  FAPB (H-Br bond distance ~ 2.37 Å, σπcalculated = 4.2 eV). Thus, we suggest that the average strength of hydrogen-bonding strongly influences the average magnitude of the σ-π splitting. Organic A-cation rotation/oscillation rates may not strongly influence the magnitude of σ-π splitting. Methylammonium and formamidinium have comparable rotation/oscillation time constants and ionic radii (rMA+ = 2.70 Å, rFA+ = 2.79 Å), but FA + hydrogen-bonds more strongly to the halide framework 26,28 . Since the σ-π splitting is a persistent/time-averaged feature in the conduction band, it is expected to influence all electron dynamics (e.g. hot electron cooling rate, potential polaron formation), irrespective of their timescale(s).
Supplementary Discussion 2. Potential connection between the σ-π character of the conduction band and polaron-like transport.
Polaron formation, involving Coulomb screening of carriers by the Br-Pb sublattice against carriercarrier, lattice defect and optical phonon scattering, has been invoked to explain the optoelectronic properties of HaPs 31 . We suggest that the σ states, having Br-Pb character, are delocalized/Bloch-like states while the π states, having Br-(A-cation) character, are spatially localized states. As femtosecondtimescale charge transfer between the organic A-cation and inorganic sublattice in the unoccupied states has been reported in MAPI (related compound), a timescale that could outcompete loss processes, we speculate that a non-thermalized electron undergoing transport will encounter both σ and π states and exhibit polaronic-like behavior (i.e. alternating between possibly lossless "trapping" and "detrapping") 32 . While occupying a localized π state, electrons may be "protected"/"screened" from scattering with electrons occupying σ states and lattice defects. The question of whether polaron dynamics can fully or partially explain the carrier dynamics of HaPs remains open 33 .